Russian Physics Journal

, Volume 59, Issue 8, pp 1153–1163 | Cite as

Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups

  • A. I. Breev
  • E. A. Mosman

The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.


Dirac equation noncommutative integration symmetry algebra 


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  1. 1.
    A. Z. Petrov, Einstein Spaces [in Russian], Fizmatlit, Moscow (1961).MATHGoogle Scholar
  2. 2.
    E. G. Kalnins, Separation of Variables for Riemannian spaces of Constant Curvature, Longman Scientific & Technical, London (1986).MATHGoogle Scholar
  3. 3.
    V. G. Bagrov and G. F. Karavaev, Exact Solutions of Relativistic Wave Equations [in Russian], Nauka, Novosibirsk (1982).Google Scholar
  4. 4.
    V. G. Bagrov and D. Gitman, The Dirac Equation and Its Solutions, De Gruyter, Berlin (2014).CrossRefMATHGoogle Scholar
  5. 5.
    V. V. Obukhov, The Steckel Spaces in Gravitation Theory [in Russian], Tomsk State Pedagogical University Publishing House, Tomsk (2006).Google Scholar
  6. 6.
    V. N. Shapovalov and G. G. Ékle, Algebraic Properties of the Dirac Equation [in Russian], Kalmyk University Publishing House, Élista (1972).Google Scholar
  7. 7.
    V. N. Shapovalov and I. V. Shirokov, Teor. Mat. Fiz., 104, No. 2, 195–213 (1995).MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. N. Shapovalov and I. V. Shirokov, Teor. Mat. Fiz., 106, No. 3–15 (1996).Google Scholar
  9. 9.
    O. L. Varaksin and I. V. Shirokov, Russ. Phys. J., 38, No. 1, 27–32 (1996).MathSciNetCrossRefGoogle Scholar
  10. 10.
    O. L. Varaksin and V. V. Klishevich, Russ. Phys. J., 39, No. 8, 727–731 (1997).MathSciNetCrossRefGoogle Scholar
  11. 11.
    V. N. Shapovalov, Sov. Phys. J., 18, No. 6, 797–802 (1975).MathSciNetCrossRefGoogle Scholar
  12. 12.
    B. Carter and R. G. McLenaghan, Phys. Rev., D19, 1093 (1979).ADSMathSciNetGoogle Scholar
  13. 13.
    R. G. McLenaghan and Ph. Spindel, Phys. Rev., D20, 409 (1979).ADSMathSciNetGoogle Scholar
  14. 14.
    R. G. McLenaghan and Ph. Spindel, Bull. Soc. Math. Belgique, XXXI, 65 (1979).Google Scholar
  15. 15.
    B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1984).MATHGoogle Scholar
  16. 16.
    S. Tachibana, Tohoku Math. J., 20, 257–264 (1968).MathSciNetCrossRefGoogle Scholar
  17. 17.
    V. V. Klishevich, Russ. Phys. J., 42, No. 10, 887–891 (2000).MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. I. Breev, I. V. Shirokov, and D. N. Razumov, Russ. Phys. J., 50, No. 10, 1012–1020 (2007).CrossRefGoogle Scholar
  19. 19.
    A. I. Breev and A. V. Kozlov, Russ. Phys. J., 58, No. 9, 1248–1257 (2015).CrossRefGoogle Scholar
  20. 20.
    A. I. Breev, Russ. Phys. J., 57, No. 8, 1050–1058 (2014).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.National Research Tomsk State UniversityTomskRussia
  2. 2.National Research Tomsk Polytechnic UniversityTomskRussia

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